Question

find tw.
tw =
submit

Explanation:

Step1: Identify Midline Theorem

The segment \( U X \) is a midline (since it connects midpoints of two sides), so \( U \) divides \( T V \) proportionally, and \( X \) divides \( T W \) proportionally. Let \( T U = x \), then \( T V = T U + U V = x + 12 \). From the left side (vertical segment of length 30, assuming \( T V \) is part of a triangle with a parallel side), wait, actually, the midline theorem (or basic proportionality theorem) implies \( \frac{T U}{U V}=\frac{T X}{X W} \). Wait, no, the midpoints: the red marks show that \( U X \) and \( V W \) have midpoints, so \( U \) is the midpoint of \( T V \)? Wait, no, the length from \( T \) to the vertical segment is 30? Wait, maybe the left side is a vertical segment of length 30, and \( T V \) has length such that \( U V = 12 \), and \( U \) is a point on \( T V \) with \( T U \) and \( U V \) in ratio. Wait, actually, the key is that \( U X \parallel V W \) (by midline, since the segments are marked as midpoints), so by the Basic Proportionality Theorem (Thales' theorem), \( \frac{T U}{U V}=\frac{T X}{X W} \). But also, the left vertical segment is 30, maybe \( T V \) is related? Wait, no, looking at the diagram, \( X W = 28 \), and we need to find \( T W = T X + X W \). Since \( U X \) is a midline, \( T X = X W \)? Wait, no, the midline connects midpoints, so if \( U \) is the midpoint of \( T V \), then \( X \) is the midpoint of \( T W \). Wait, but \( U V = 12 \), so \( T U = 12 \) (if \( U \) is midpoint)? No, wait, the left vertical segment is 30, maybe \( T V \) is 30? Wait, the left side has a vertical segment of length 30, labeled with \( T \) at the bottom and the top at the horizontal line. So \( T V \) is a side of the triangle, with \( U \) on \( T V \) and \( U V = 12 \), \( T U = 30 - 12 = 18 \)? No, that doesn't make sense. Wait, maybe the vertical segment is \( T V \) with length 30? Wait, the diagram shows a vertical segment with length 30, and \( T V \) has \( U \) such that \( U V = 12 \), so \( T U = 30 - 12 = 18 \)? Then, by the Basic Proportionality Theorem, since \( U X \parallel V W \), \( \frac{T U}{U V}=\frac{T X}{X W} \). So \( \frac{18}{12}=\frac{T X}{28} \). Solving for \( T X \): \( T X=\frac{18\times28}{12}=42 \)? No, that can't be. Wait, maybe \( T V \) is 30, and \( U V = 12 \), so \( T U = 30 - 12 = 18 \), but the ratio is \( T U:U V = 18:12 = 3:2 \). Then \( T X:X W = 3:2 \), but \( X W = 28 \), so \( T X=\frac{3}{2}\times28 = 42 \), then \( T W = T X + X W = 42 + 28 = 70 \)? Wait, no, maybe the midline is such that \( U \) is the midpoint, so \( T U = U V = 12 \), but then the vertical segment would be 24, not 30. Wait, the vertical segment is 30, so \( T V = 30 \), \( U V = 12 \), so \( T U = 30 - 12 = 18 \). Then, since \( U X \parallel V W \), \( \frac{T U}{T V}=\frac{T X}{T W} \). Wait, \( T V = T U + U V = 18 + 12 = 30 \), so \( \frac{18}{30}=\frac{T X}{T X + 28} \). Cross-multiplying: \( 18(T X + 28)=30 T X \) → \( 18 T X + 504 = 30 T X \) → \( 12 T X = 504 \) → \( T X = 42 \). Then \( T W = T X + X W = 42 + 28 = 70 \). Wait, but maybe the correct approach is that \( U \) is the midpoint, so \( T U = U V \), but the vertical segment is 30, so \( T V = 30 \), \( U V = 12 \), so that's not midpoint. Wait, maybe the diagram has a typo, but the standard problem: if \( U X \) is a midline, then \( T X = X W \), but \( X W = 28 \), so \( T X = 28 \), \( T W = 56 \)? No, that contradicts. Wait, no, the key is that the left vertical segment is 30, and \( T V \) is 30, \( U V = 12 \), so \( T U = 18 \), ratio \( 18:12 = 3:2 \), so \( T X:X W = 3:2 \), \( X W = 28 \), so \( T X = (3/2)*28 = 42 \), \( T W = 42 + 28 = 70 \).

Step2: Calculate \( T W \)

Using the ratio from the Basic Proportionality Theorem (Thales' theorem) because \( U X \parallel V W \), so \( \frac{T U}{U V}=\frac{T X}{X W} \). Let \( T U = 30 - 12 = 18 \) (assuming the vertical segment is \( T V = 30 \)), \( U V = 12 \), \( X W = 28 \). Then \( \frac{18}{12}=\frac{T X}{28} \) → \( T X = \frac{18 \times 28}{12} = 42 \). Then \( T W = T X + X W = 42 + 28 = 70 \).

Answer:

\( 70 \)